Step 1: What they are, in Maxwell's picture.
Maxwell showed that an accelerating charge sets up a time-varying electric field, which by his equations creates a time-varying magnetic field, which in turn feeds back into an electric field. This self-supporting chain of oscillating fields spreading outward is an electromagnetic wave; no vibrating particles of a medium are involved, only the fields themselves.
Step 2: Characteristic 1, no medium and transverse nature.
Because the wave is built out of \(\vec{E}\) and \(\vec{B}\) that create each other, it does not rely on air, water or any material to travel and moves happily through vacuum. Both fields vibrate at right angles to the direction the wave moves and at right angles to each other, so the wave is transverse. This transverse character is what makes light capable of being polarised.
Step 3: Characteristic 2, universal speed and energy.
In vacuum every electromagnetic wave, from radio waves to gamma rays, travels at one fixed speed set only by the electric and magnetic constants of space:
\[c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3\times10^{8}\ \text{m/s},\qquad \frac{E_0}{B_0} = c\]
The wave also transports energy shared equally between its electric and magnetic fields and exerts radiation pressure because it carries momentum.
Step 4: Conclusion.
Thus an electromagnetic wave is a medium-free, transverse wave of coupled electric and magnetic fields moving at the speed of light and carrying energy.
\[\boxed{\vec{E}\perp\vec{B}\perp\text{propagation};\ \ c=1/\sqrt{\mu_0\varepsilon_0}}\]